MEASURING FRACTALS BY INFINITE AND INFINITESIMAL NUMBERS Yaroslav D. Sergeyev DEIS, University of Calabria, Via P. Bucci, Cubo 42-C, 87036 Rende (CS), Italy, N.I. Lobachevsky State University, Nizhni Novgorod, Russia, and Institute of High Performance Computing and Networking of the National Research Council of Italy http://wwwinfo.deis.unical.it/∼yaro e-mail: [email protected] Abstract. Traditional mathematical tools used for analysis of fractals allow one to distinguish results of self-similarity processes after a finite number of iterations. For example, the result of procedure of construction of Cantor’s set after two steps is different from that obtained after three steps. However, we are not able to make such a distinction at infinity. It is shown in this paper that infinite and infinitesimal numbers proposed recently allow one to measure results of fractal processes at different iterations at infinity too. First, the new technique is used to measure at infinity sets being results of Cantor’s proce- dure. Second, it is applied to calculate the lengths of polygonal geometric spirals at different points of infinity. 1. INTRODUCTION During last decades fractals have been intensively studied and applied in various fields (see, for instance, [4, 11, 5, 7, 12, 20]). However, their mathematical analysis (except, of course, a very well developed theory of fractal dimensions) very often continues to have mainly a qualitative character and there are no many tools for a quantitative analysis of their behavior after execution of infinitely many steps of a self-similarity process of construction. Usually, we can measure fractals in a way and can give certain numerical answers to questions regarding fractals only if a finite number of steps in the procedure of their construction has been executed. The same questions can remain without any answer if we consider execution of an infinite number of steps. For example, let us consider the famous fractal construction – Cantor’s set (see Fig. 1). If a finite number of steps, n, has been done constructing Cantor’s set, then we are able to describe numerically the set being the result of this operation. It will have 2n intervals having the length 0Received March 16, 2006. Revised September 12, 2006. 02000 Mathematics Subject Classification: 28A80, 40A05, 40G99, 03E99. 0Keywords: Fractals, infinite and infinitesimal numbers, numeral systems. 2 Yaroslav D. Sergeyev Step 0 Step 1 Step 2 Step 3 Step 4 FIGURE 1. Cantor’s construction. 1 3n each. Obviously, the set obtained after n + 1 iterations will be different and we also are able to measure the lengths of the intervals forming the second set. It will n+1 1 have 2 intervals having the length 3n+1 each. The situation changes drastically in the limit because we are not able to distinguish results of n and n + 1 steps of the construction if n is infinite. We also are not able to distinguish at infinity the results of the following two processes that both use Cantor’s construction but start from different positions. The first one is the usual Cantor’s set and it starts from the interval [0;1], the second starts 1 2 from the couple of intervals [0; 3 ] and [ 3 ;1]. In spite of the fact that for any given finite number of steps, n, the results of the constructions will be different for these two processes, we have no tools to distinguish and, therefore, to measure them at infinity. Another class of fractal objects that defies length measurement are spirals that fas- cinated mathematicians throughout the ages (see, e.g., [12]). Let us consider two kinds of polygonal spirals shown in Figs. 2 and 3. Both of them are geometric polyg- onal spirals related to geometric sequences. The spiral shown in Fig. 2 is constructed as follows. The unit interval is our initial piece and we draw it vertically from bottom to top. At the end we make a right turn and draw the unit interval again from left to right. Then we draw the interval having a length q < 1 by continuation in the same direction from left to right. At the end we make another right turn and draw again the same interval having the length q from top to bottom. At the end of this line we draw the interval with the length q2 and continue using the same principle. Fig. 3 shows the same construction for q > 1. Evidently, for q = 1 we obtain just a square. If we try to calculate the length of geometric polygonal spirals, we obtain imme- diately that it is equal to ∞ S = 2(1 + q + q2 + q3 + :::) = 2 ∑ qi; (1.1) i=0 which is a geometric series and, therefore, its limiting length for q < 1 is 2=(1 − q), i.e., a finite value different for each given q. Then, for all q > 1 traditional analysis Measuring fractals by infinite and infinitesimal numbers 3 FIGURE 2. The first construction steps of a polygonal geometric spi- ral with q < 1. FIGURE 3. The first construction steps of a polygonal geometric spi- ral with q > 1. tells us that the spiral has the infinite length, i.e., we are not able to distinguish the spirals in dependence of the value of q. In this paper, we show how a recently developed approach (see [8, 15, 16, 17, 18, 19]) that allows one to write down infinite and infinitesimal numbers and to execute arithmetical operations with them can be used for measuring fractals at infinity. Par- ticularly, the lengths of intervals of Cantor’s set and the lengths of spirals from Figs. 2 and 3 for any q will be calculated. The rest of the paper is organized as follows. Section 2 introduces the new method- ology and Section 3 describes a general framework allowing one to express by a finite number of symbols not only finite but infinite and infinitesimal numbers, too. Sec- tion 4 describes how infinite and infinitesimal numbers can be used for measuring fractal objects. Finally, Section 5 contains a brief conclusion. 4 Yaroslav D. Sergeyev 2. METHODOLOGY Usually, when mathematicians deal with infinite objects (sets or processes) it is supposed that human beings are able to execute certain operations infinitely many times (see [1, 2, 3, 10, 14]). For example, in a fixed numeral system it is possible to write down a numeral1 with any number of digits. However, this supposition is an abstraction (courageously declared by constructivists in [9]) because we live in a finite world and all human beings and/or computers finish operations they have started. The new computational paradigm introduced in [16, 17, 18, 19] does not use this abstraction and, therefore, is closer to the world of practical calculations than tra- ditional approaches. Its strong computational character is enforced also by the fact that the first simulator of the Infinity Computer able to work with infinite, finite, and infinitesimal numbers introduced in [16, 17, 18, 19] has been already realized (see [8, 15]). In order to introduce the new methodology, let us consider a study published in Science by Peter Gordon (see [6]) where he describes a primitive tribe living in Ama- zonia - Piraha˜ - that uses a very simple numeral system for counting: one, two, many. For Piraha,˜ all quantities bigger than two are just ‘many’ and such operations as 2+2 and 2+1 give the same result, i.e., ‘many’. Using their weak numeral system Piraha˜ are not able to see, for instance, numbers 3, 4, 5, and 6, to execute arithmetical op- erations with them, and, in general, to say anything about these numbers because in their language there are neither words nor concepts for that. Moreover, the weakness of their numeral system leads to such results as ‘many’ + 1 = ‘many’; ‘many’ + 2 = ‘many’; which are very familiar to us in the context of views on infinity used in the traditional calculus ∞ + 1 = ∞; ∞ + 2 = ∞: This observation leads us to the following idea: Probably our difficulty in working with infinity is not connected to the nature of infinity but is a result of inadequate numeral systems used to express numbers. We start by introducing three postulates that will fix our methodological positions with respect to infinite and infinitesimal quantities and mathematics, in general. Postulate 1. We accept that human beings and machines are able to execute only a finite number of operations. 1We remind that numeral is a symbol or group of symbols that represents a number. The difference between numerals and numbers is the same as the difference between words and the things they refer to. A number is a concept that a numeral expresses. The same number can be represented by different numerals. For example, the symbols ‘6’, ‘six’, and ‘VI’ are different numerals, but they all represent the same number. Measuring fractals by infinite and infinitesimal numbers 5 Thus, we accept that we shall never be able to give a complete description of infinite processes and sets due to our finite capabilities. Particularly, this means that we accept that we are able to write down only a finite number of symbols to express numbers. The second postulate that will be adopted is due to the following consideration. In natural sciences, researchers use tools to describe the object of their study and the used instrument influences results of observations. When physicists see a black dot in their microscope they cannot say: The object of observation is the black dot. They are obliged to say: the lens used in the microscope allows us to see the black dot and it is not possible to say anything more about the nature of the object of observation until we’ll not change the instrument - the lens or the microscope itself - by a more precise one.